Applications of Multiple Integrals

Double Integrals

Quantity General Formula In Cartesian Coordinates In Polar Coordinates
Area of a planar region S=∬RdAS = \iint_R dA ∬Rdx dy\displaystyle \iint_R dx\,dy ∬Rρ dρ dφ\displaystyle \iint_R \rho\,d\rho\,d\varphi
Surface area of z=f(x,y)z = f(x,y) S=∬R1+(∂z∂x)2+(∂z∂y)2 dAS = \iint_R \sqrt{1 + \left( \frac{\partial z}{\partial x} \right)^2 + \left( \frac{\partial z}{\partial y} \right)^2} \,dA ∬R1+fx2+fy2 dx dy\displaystyle \iint_R \sqrt{1 + f_x^2 + f_y^2}\,dx\,dy ∬Rρ2+ρ2fρ2+fφ2 dρ dφ\displaystyle \iint_R \sqrt{\rho^2 + \rho^2 f_\rho^2 + f_\varphi^2}\,d\rho\,d\varphi
Volume under a surface V=∬Rz dAV = \iint_R z\,dA ∬Rf(x,y) dx dy\displaystyle \iint_R f(x,y)\,dx\,dy ∬Rz(ρ,φ) ρ dρ dφ\displaystyle \iint_R z(\rho,\varphi)\,\rho\,d\rho\,d\varphi
Moment of inertia about the (x)-axis Ix=∬Ry2 dAI_x = \iint_R y^2\,dA ∬Ry2 dx dy\displaystyle \iint_R y^2\,dx\,dy ∬Rρ3sin⁡2φ dρ dφ\displaystyle \iint_R \rho^3\sin^2\varphi\,d\rho\,d\varphi
Moment of inertia about the origin (O) IO=∬R(x2+y2) dAI_O = \iint_R (x^2+y^2)\,dA ∬R(x2+y2) dx dy\displaystyle \iint_R (x^2+y^2)\,dx\,dy ∬Rρ3 dρ dφ\displaystyle \iint_R \rho^3\,d\rho\,d\varphi
Mass of a lamina with density δ(x,y)\delta(x,y) M=∬Rδ dAM = \iint_R \delta\,dA ∬Rδ(x,y) dx dy\displaystyle \iint_R \delta(x,y)\,dx\,dy ∬Rδ(ρ,φ) ρ dρ dφ\displaystyle \iint_R \delta(\rho,\varphi)\,\rho\,d\rho\,d\varphi
Center of mass coordinates (homogeneous lamina) {xc=1S∬Rx dAyc=1S∬Ry dA\begin{cases} \displaystyle x_c = \frac{1}{S}\iint_R x\,dA \\[1em] \displaystyle y_c = \frac{1}{S}\iint_R y\,dA \end{cases} {∬Rx dx dy∬Rdx dy∬Ry dx dy∬Rdx dy\begin{cases} \displaystyle \frac{\iint_R x\,dx\,dy}{\iint_R dx\,dy} \\[1em] \displaystyle \frac{\iint_R y\,dx\,dy}{\iint_R dx\,dy} \end{cases} {∬Rρ2cos⁡φ dρ dφ∬Rρ dρ dφ∬Rρ2sin⁡φ dρ dφ∬Rρ dρ dφ\begin{cases} \displaystyle \frac{\iint_R \rho^2\cos\varphi\,d\rho\,d\varphi}{\iint_R \rho\,d\rho\,d\varphi} \\[1em] \displaystyle \frac{\iint_R \rho^2\sin\varphi\,d\rho\,d\varphi}{\iint_R \rho\,d\rho\,d\varphi} \end{cases} 
title: Notes
- $dA$: differential area element.
- For surface area, $\gamma$ is the angle between the surface normal and the $z$-axis.
- In polar coordinates: $x = \rho\cos\varphi$, $y = \rho\sin\varphi$, $dA = \rho\,d\rho\,d\varphi$.
- For homogeneous laminas, the constant density $\delta$ cancels out in center of mass formulas.

Triple Integrals

Quantity General Formula Cartesian Coordinates Cylindrical Coordinates Spherical Coordinates
Volume of a solid V=∭VdVV = \iiint_V dV ∭Vdx dy dz\displaystyle \iiint_V dx\,dy\,dz ∭Vρ dρ dφ dz\displaystyle \iiint_V \rho\,d\rho\,d\varphi\,dz ∭Vr2sin⁡θ dr dθ dφ\displaystyle \iiint_V r^2\sin\theta\,dr\,d\theta\,d\varphi
Moment of inertia about the (z)-axis Iz=∭V(x2+y2) dVI_z = \iiint_V (x^2+y^2)\,dV ∭V(x2+y2) dx dy dz\displaystyle \iiint_V (x^2+y^2)\,dx\,dy\,dz ∭Vρ3 dρ dφ dz\displaystyle \iiint_V \rho^3\,d\rho\,d\varphi\,dz ∭Vr4sin⁡3θ dr dθ dφ\displaystyle \iiint_V r^4\sin^3\theta\,dr\,d\theta\,d\varphi
Mass of a solid with density δ(x,y,z)\delta(x,y,z) M=∭Vδ dVM = \iiint_V \delta\,dV ∭Vδ(x,y,z) dx dy dz\displaystyle \iiint_V \delta(x,y,z)\,dx\,dy\,dz ∭Vδ(ρ,φ,z) ρ dρ dφ dz\displaystyle \iiint_V \delta(\rho,\varphi,z)\,\rho\,d\rho\,d\varphi\,dz ∭Vδ(r,θ,φ) r2sin⁡θ dr dθ dφ\displaystyle \iiint_V \delta(r,\theta,\varphi)\,r^2\sin\theta\,dr\,d\theta\,d\varphi
Center of mass coordinates (homogeneous solid) {xc=1V∭Vx dVyc=1V∭Vy dVzc=1V∭Vz dV\begin{cases} \displaystyle x_c = \frac{1}{V}\iiint_V x\,dV \\[1em] \displaystyle y_c = \frac{1}{V}\iiint_V y\,dV \\[1em] \displaystyle z_c = \frac{1}{V}\iiint_V z\,dV \end{cases} {∭Vx dx dy dz∭Vdx dy dz∭Vy dx dy dz∭Vdx dy dz∭Vz dx dy dz∭Vdx dy dz\begin{cases} \displaystyle \frac{\iiint_V x\,dx\,dy\,dz}{\iiint_V dx\,dy\,dz} \\[1em] \displaystyle \frac{\iiint_V y\,dx\,dy\,dz}{\iiint_V dx\,dy\,dz} \\[1em] \displaystyle \frac{\iiint_V z\,dx\,dy\,dz}{\iiint_V dx\,dy\,dz} \end{cases} {∭Vρ2cos⁡φ dρ dφ dz∭Vρ dρ dφ dz∭Vρ2sin⁡φ dρ dφ dz∭Vρ dρ dφ dz∭Vzρ dρ dφ dz∭Vρ dρ dφ dz\begin{cases} \displaystyle \frac{\iiint_V \rho^2\cos\varphi\,d\rho\,d\varphi\,dz}{\iiint_V \rho\,d\rho\,d\varphi\,dz} \\[1em] \displaystyle \frac{\iiint_V \rho^2\sin\varphi\,d\rho\,d\varphi\,dz}{\iiint_V \rho\,d\rho\,d\varphi\,dz} \\[1em] \displaystyle \frac{\iiint_V z\rho\,d\rho\,d\varphi\,dz}{\iiint_V \rho\,d\rho\,d\varphi\,dz} \end{cases} {∭Vr3sin⁡2θcos⁡φ dr dθ dφ∭Vr2sin⁡θ dr dθ dφ∭Vr3sin⁡2θsin⁡φ dr dθ dφ∭Vr2sin⁡θ dr dθ dφ∭Vr3sin⁡θcos⁡θ dr dθ dφ∭Vr2sin⁡θ dr dθ dφ\begin{cases} \displaystyle \frac{\iiint_V r^3\sin^2\theta\cos\varphi\,dr\,d\theta\,d\varphi}{\iiint_V r^2\sin\theta\,dr\,d\theta\,d\varphi} \\[1em] \displaystyle \frac{\iiint_V r^3\sin^2\theta\sin\varphi\,dr\,d\theta\,d\varphi}{\iiint_V r^2\sin\theta\,dr\,d\theta\,d\varphi} \\[1em] \displaystyle \frac{\iiint_V r^3\sin\theta\cos\theta\,dr\,d\theta\,d\varphi}{\iiint_V r^2\sin\theta\,dr\,d\theta\,d\varphi} \end{cases} 
title: Notes
- **Cylindrical coordinates**: $x = \rho\cos\varphi$, $y = \rho\sin\varphi$, $z = z$, $dV = \rho\,d\rho\,d\varphi\,dz$.
- **Spherical coordinates**: $x = r\sin\theta\cos\varphi$, $y = r\sin\theta\sin\varphi$, $z = r\cos\theta$, $dV = r^2\sin\theta\,dr\,d\theta\,d\varphi$.
- For homogeneous solids, the constant density $\delta$ cancels out in center of mass formulas.
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